On torsion freeness for the decomposable Orlik-Solomon algebra

We prove the torsion freeness of the decomposable Orlik–Solomon algebra of a simple matroid on ground set $[n]$. In the class of hypersolvable & non-supersolvable complex hyperplane arrangements, the torsion freeness, in a certain degree, of this combinatorially defined object, associated to the intersection lattice of the arrangement, impacts on the first non-vanishing higher homotopy group of the complement of the arrangement.